To find the kinetic and potential energy at the top and bottom of each hill, we need to use the formulas for kinetic energy (KE) and potential energy (PE). The formulas are:
KE = (1/2) * m * v^2
PE = m * g * h
Where:
KE is the kinetic energy
m is the mass of the roller coaster car (800.0 kg)
v is the velocity of the roller coaster car
PE is the potential energy
g is the acceleration due to gravity (9.8 m/s^2)
h is the height of the roller coaster car
Let's calculate the kinetic and potential energy at each point:
1. Top of the first hill (initial position):
At the top of the hill, the roller coaster car is at rest, so its velocity is 0.
KE = (1/2) * 800.0 kg * 0^2 = 0 J
PE = 800.0 kg * 9.8 m/s^2 * 95 m = 752,800 J
2. Bottom of the first hill:
The roller coaster car reaches a height of 31 m at the bottom of the first hill.
KE = (1/2) * 800.0 kg * v^2
PE = 800.0 kg * 9.8 m/s^2 * 31 m
To find the velocity (v) at the bottom of the hill, we can use the conservation of energy principle, which states that the total mechanical energy (the sum of kinetic and potential energy) remains constant. Therefore, the initial potential energy at the top is equal to the final kinetic energy at the bottom:
KE (bottom) = PE (top)
(1/2) * 800.0 kg * v^2 = 752,800 J
Solving for v, we find:
v = sqrt(2 * (752,800 J) / (800.0 kg)) â 30.05 m/s
So,
KE (bottom) = (1/2) * 800.0 kg * (30.05 m/s)^2 â 360,563 J
PE (bottom) = 800.0 kg * 9.8 m/s^2 * 31 m â 241,360 J
3. Top of the second hill:
The roller coaster car reaches a height of 0 m at the top of the second hill and is moving with a velocity of 28 m/s.
KE = (1/2) * 800.0 kg * (28 m/s)^2 â 313,600 J
PE = 800.0 kg * 9.8 m/s^2 * 0 m = 0 J
4. Bottom of the second hill (ground level):
The roller coaster car reaches a height of 0 m at the bottom of the second hill.
KE = (1/2) * 800.0 kg * v^2
PE = 800.0 kg * 9.8 m/s^2 * 0 m
Using the same conservation of energy principle as before:
KE (bottom) = (1/2) * 800.0 kg * (28 m/s)^2 â 313,600 J
PE (bottom) = 800.0 kg * 9.8 m/s^2 * 0 m = 0 J
So, the kinetic and potential energy at each point are:
First hill:
Top: KE = 0 J, PE â 752,800 J
Bottom: KE â 360,563 J, PE â 241,360 J
Second hill:
Top: KE â 313,600 J, PE = 0 J
Bottom: KE â 313,600 J, PE = 0 J